Banach Algebras
In these notes we give the main theory of Banach algebras, restricting the discussion to the clearest case—commutative Banach algebras with identity. This allows for an uncluttered presentation of the main ideas. Generalizations and refinements can then be studied with the main principles already at hand.
Let's briefly discuss the purpose of this study. It's easy to prove that the reciprocal of a nonvanishing continuous function on the unit interval is a continuous function on the unit interval. This can be proved directly by showing that the reciprocal satisfies the definition of continuity. But here's an immeasureably harder problem. Norbert Wiener's theorem on absolutely convergent trigonometric series: The reciprocal of a nonvanishing absolutely convergent trigonometric series is an absolutely convergent trigonometric series. Precisely,
∞ 38> let 0> œ +/8 be an absolutely convergent trigonometric series. If 0> Á! 8œ∞ " a> − !ß #1 , then 0> is an absolutely convergent trigonometric series. c Wiener proved this theorem analytically, by basically inverting the series. His proof is long and difficult. But we will see that we can prove Wiener's theorem as a simple consequence of Banach Algebra theorems (a result first proved by Gelfond). Gelfond shows that the reciprocal of 0 is an absolutely convergent trigonometric series by showing that 0 belongs to a collection of functions that have this property. Gelfond's proof using Banach Algebras is so simple and elegant that it inspired mathematicians to find other uses for this "functional analysis" approach (that is, looking at algebraic collections of functions instead of working with single functions).
0 1 0 1 Analysis Functional Analysis Properties of 0GÒ!ß"Ó Properties of
1 0. Definitions
Definition: An algebra is a vector space E with addition (+) and scalar multiplication (juxtaposition) with an additional operation called multiplication (also juxtaposition) with the following properties:
" BC D œB CD aBßCßD−E Ð#Ñ B C D œ BC BD aBß Cß D − E CD BœCBDB aBßCßD−E Ð$Ñαα BC œ B C œ B α C a α‚ − à Bß C − E An algebra E is called commutative if Ð%Ñ BC œ CB aBß C − E
An algebra E has an identity if b/−E such that /BœB/œB for all B−EÞ A subalgebra FE of an algebra is a subset of E which is closed under addition and multiplication.
Definition: A normed algebra is an algebra E†E߆ with a norm such that is a ll ll normed vector space (i.e. †EÄÑ : ‘ with: ll (1) Bœ!iff Bœ! ll (2) ααBœ B ll kkll Ð$Ñ B C Ÿ B C (which means is continuous) llllll with the additional property that
(4) BC Ÿ B C (which means † is continuous) ll llll
1.1 Theorem: Let E be a normed algebra. Then multiplication is continuous from EEÄE× (i.e. jointly continuous).
Proof - Suppose BÄB8 and CÄC8 We need to show that BC88 ÄBC .
B C BC œ B C B CB CBC lll888888 l ŸBCBCBCBC llll8 88 8 œBCC BBC llll8 88 ŸB CCBBC lll88 l l8 lll ŸQ C C B B C (since B converges) llllll88 8 Ä!
Definition: A Banach algebra is a complete normed algebra.
2 Definition: A homomorphism from an algebra EF to an algebra is a linear map 9 satisfying 999ÐB CÑ œ ÐBÑ ÐCÑ 999ÐBCÑ œ ÐBÑ ÐCÑ
A bijective homomorphism 9 from EF onto is called an algebra isomorphism . The normed algebras EF and are isomorphic (as normed algebras) if there is an algebra isomorphism 9 ÀEÄF which preserves the norm
9ÐBÑ œ B llll In this case EF is identical to as a normed algebra (the isomorphism merely renames the elements of the algebra). If EF is isomorphic with a subalgebra of by a 9 then we say that EF is embedded in and 9 is called an embedding.
Exercises
1. The center of an algebra EG is the subset consisting of those elements that commute with every element in EE . The center is a commutative subalgebra of . 2. If E is a Banach algebra with norm † and identity / , then the norm l†l defined by " ll lBl œ/ B is a vector space norm on A which is equivalent to the original norm. The llll norm lBl is not necesarily a Banach Algebra norm (i.e, not necessarily submultiplicative.). 3. Show that the algebras E œ GÒ"ß #Ó and F œ GÒ$ß 4 Ó are isomorphic Banach algebras In each case the algebras are equipped with pointwise operations and the sup norm.
3 1. Examples and Constructions
In this section we give some fundamental examples of Banach algebras. We also define some basic constructions including products, quotients, and direct sums of Banach algebras. Other are presented as needed.
Example of Banach Algebras
There are three general categories of Banach algebras: "algebras of continuous functions," "algebras of operators," and "group algebras."
Example 1. Algebras of Continuous Functions Let GÒ 0,1] œ the set of continuous complex-valued functions on !ß" , where operations are defined pointwise. The norm is cd
Bœsup B> ll>−Ò!ß"Ó k k
It's easy to show that BCŸB C and BCŸBC . So, with this norm l l llll l l ll ll GÒ!ß "Ó is a commutative Banach algebra with identity / > ´ "Þ A Note on GÐ\Ñ The Banach algebra GÒ!ß "Ó is a special case of the following more general construction. Let \GÐ\Ñ be a compact topological space and be the collection of real (or complex) valued continuous functions on \GÐ\Ñ . is a Banach algebra with pointwise operations and the norm Bœsup B> ll>−\ k k
If \œÒ!ß"Ó then GÐ\Ñ is the algebra in Example 1.
Example 2. Algebras of Analytic Functions Let EH œ the analytic functions on the interior of Hœ D D Ÿ" and continuous on ˜™¸ kk the boundary “ œDDœ"Þ Operations are +, † and scalar multiplication. ˜™¸kk The norm is
0œ sup 0> ll>−H k k œ0>sup (by the Maximum Modulus Principle) >−“ kk
Example 3. Algebras of Operators Let \F\ be a Banach space. TT = , i.e. is the collection of all bounded linear operators on \F\ . We know is a Banach space (since \ is Banach). Multiplication is given by functional composition:
4 XWBœXWB aB−\ Also, XW Ÿ X W Þ llllll Thus, F\ is a Banach algebra with identity M , where MB œB for all B−\
Example 4. Algebras of Integrable Functions Let Pœ" ‘‘ the set of all equivalent classes of integrable functions on . i.e. 0 œ 0 .. ∞ , where 0µ1 iff 0œ1 except possible on a set of ll'‘ kk measure zero, and the integral is the Lebesgue integral. P" ‘ is a vector space under addition and scalar multiplication with the norm 0œ 0.Þ. ll'‘ kk We have:
a) 0œ!0œ! iff a.e. (almost everywhere) ll b) α.α0. œ 0. . means α 0 œ α 0 ''‘‘k k kk kk l l kkll -Ñ 01.... Ÿ 0. 1.means 01 Ÿ 0 1 '''‘kk‘‘ kkkk llllll ¾P" ‘ is a Banach space Multiplication is given by convolution:
0‡1 B œ 0BC1C.C '‘ Need to show (‡ ) is associative, distributive, compatible with multiplication, and
0‡1 Ÿ 0 1 Þ llllll Proof - 0‡1 œ∞ 0‡1 B .B ll'∞ k k œ0BC1C.C.B∞∞ ''∞kk ∞ Ÿ0BC1C.C.B∞∞ ''∞ ∞kk œ0BC1C.B.C∞∞ (Fubini's Theorem) ''∞ ∞kk œ1C0BC.B.C∞∞ ''∞kkk ∞ k œ∞∞ 1C 0B.B.C ∞ 0B.B−‘ ''∞kkkk ∞ ' ∞ k k œ0B.B1C.C∞∞ ''∞kk ∞ kk œ0 1 llll ¾0‡1Ÿ0 1 llllll
5 Now, in the expression 0‡1 B œ∞ 0 BC1C .C '∞ let ? œ B C Ê .? œ .C and C œ B ? . Then,
0‡1 B œ ∞ 0 ?1B? .? '∞ œ1B?0C.?∞ '∞ œ1‡0B So multiplication is commutative. There is no identity. Thus P" ‘ is a commutative Banach algebra without identity. A Note on Topological Groups PPKK""‘ is a special case of where is a topological group with Haar measure. A topological group is a group K with a topology 77 such that addition and inversion are - continuous. Every topological group has a unique translation invariant measure . called Haar measure. (Translation invariant means ..EB œ E, for E§KÑ . Examples of topological groups are
1) ‘ß with usual topology. 2) ““1ß † œ the unit circle, with complex multiplication. = /3> > − !ß # ˜™¸ ‰ $Ñ™ Î 8 with discrete topology. The algebra PÐKÑ" consistting of the the measurable functions on K satisfying
0œ l0l.∞. ll 'K is a Banach algebra under convolution (‡ )
0‡1Ð=Ñ œ 0Ð= >Ñ1Ð>Ñ .. Ð>Ñ 'K
Example 5. The Group Algebra 61 ™
∞ 1 ∞ Let 6œ+™‚8 8œ∞ +−88 and +∞ where œ ¸ 8œ∞
∞ +œ +8 ll 8œ∞ and multiplication is given by
∞∞ +‡,œ +85 , 5 Þ 8œ∞ Œ 5œ∞
6 Example 6. Wiener's Algebra The algebra [ (for Weiner) consists of the absolutely convergent trigonometric series
∞∞ 38> [œ 0> œαα88 / 0 œ ∞ œ 8œ∞ ¹ l l 8œ∞ k k
∞∞ 38> 38> with convolution as the multiplication: for 0> œα"88 / and 1> œ / 8œ∞ 8œ∞
∞∞ 38> 38> 01 > œα"88 / / Œ Œ 8œ∞ 8œ∞
∞∞ 38> œ/α"85 5 8œ∞ Œ 5œ∞
Constructions with Banach Algebras
Here we show how new Banach algebras can be constructed from other Banach algebras.
Definition: If EE"# and are Banach algebras then their direct sum is the algebra
EœE"# ŠE
consisting of ordered pairs ÐB"# ß B Ñ with coordinate wise operations and with the norm Bœmax ÐBßBÑ . ll llll"# Example 7. Dircct Sums 1. The direct sum GÒ!ß "Ó Š GÒ!ß "Ó is isomorphic to GÐÒ!ß "Ó ∪ Ò#ß $ÓÑ . 2. The direct sum -‚-!! is isomorphic to - ! .
Definition: If EME is a Banach algebras and is a proper ideal in , then the quotient algebra EÎM is a normed algebra with the norm
BM =inf BC llC−M ll
Quotient algebras are studied in more detail in Section 4. This idea is key to the development of the Gelfand theory for Banach algebras.
Example 7. Quotient Algebras 1. Let E œ P" ÐKÑ Š GÒ!ß "Ó. What is EÎGÒ!ß "Ó . 2. In general, if EœE"# ŠE , it's easy to see that EÎE " œE # .
7 Exercises
1. The collection FÐ!ß "Ñ of bounded continuous functions on Ð!ß "Ñ with pointwise operations and sup norm is a Banach algebra.
2. The Banach space 6ÐÑ∞ is a Banach algebra if multiplication is defined pointwise.
3. The Banach space PÐÑ∞ ‘ is a Banach algebra if multiplication is defined pointwise. [Note on 2 and 3: In general, if Ð\ßH. ß Ñ is a 5 -finite measure space then PÐ\ßßÑ∞ H. is a Banach algebra under pointwise multiplication.]
4. PÐ!ß∞ÑœÖB−PÐ""‘‘ ÑÀBÐ>Ñœ!ß>Ÿ!× is a closed subalgebra of PÐ " Ñ .
5. If I§Ò!ß"Ó then FœÖB−GÒ!ß"ÓÀBÐIÑœ!× is a closed subalgebra of GÒ!ß"Ó .
6. P" Ð!ß 1 Ñ œ ÖB À Bœ" BÐ>Ñ.>∞× is a Banach algebra with convolution ll'! k k B‡C œ "BÐ> 777 ÑCÐ Ñ . Ð! > "Ñ '! and the obvious norm. (This algebra has no identity).
7. P: Г Ñ œ ÖB À Bœ BÐ>Ñ.>∞×: is a Banach algebra with convolution ll'“ k k B‡C œ BÐ>777" ÑCÐ Ñ . '“ and the obvious norm. [Note on 6: If K: "PÐKÑ is a compact topological group, and , then : is a Banach algebra under convolution.]
8. Let G8 Ò!ß "Ó denote the algebra of all 8 -times differentiable functions on Ò!ß "Ó with pointwise operations. Then G8 Ò!ß "Ó is a Banach algebra with the norm
8 lBÐ5Ñ Ð>Ñl Bœ8 sup 5x ll 0t1ŸŸ 5œ!
(We use the convention that the derivatives at the endpoints are one-sided)
9. An approximate identity in EÐ/Ñ is a net α of bounded norm such that lim/Bœαα lim B/œB αα for every B−E . In PÐ" ‘ Ñ the sequence of functions
"" #8for 88 Ÿ > Ÿ /Ð>Ñœ8 " !l>lfor 8
8 is an approximate identity.
10. (a) If EE"# and are Banach algebras then EŠE "# is a Banach algebra. (b) EŠE"# has an identity if and only if E " and E # have identities.
11. GÒ!ß "Ó Š GÒ!ß "Ó is isomorphic to GÐÒ!ß "Ó ∪ Ò#ß $ÓÑ .
12. If EEEá"#, , 3 , is a sequence of Banach algebras then their bounded direct sum is the subset of the product consisting of those for which 8EBœÐBÑ88 supB∞Þ The bounded direct sum is a Banach algebra. 88ll
9 2. Basic Properties of Commutative Banach Algebras
Let E be a commutative Banach algebra over ‚ . In this section we prove that a Banach algebra without identity can be imbedded in a Banach algebra with identity. Also, in any Banach algebra with identity we can always find an equivalent norm in which the norm of the identity is 1.
2.1 Theorem: (Adjoining an Identity) A Banach algebra E without identity can always be embedded as a subalgebra of a Banach algebra with identity. (This procedure ~ is called adjoining an identity and denoted EœEŠ " ). ef Proof - Let FœEŠ‚ with the following operations:
1) Bßα" Cß œ B Cß α" 2) -Bßαα œ -Bß- 3) Bßα" Cß œ BC α"α" C Bß 4) Bßαα œ B llllkk Note that !ß " is an identity for F . The embedding is given by:
XÀEÄF where BÈ Bß! Note that X is a homomorphism of algebras and preserves the norm so that X is injective and an isometry. i.e. XBœ Bß!œB!œB l l l l ll ll ll
The next theorem shows that any Banach algebra can be represented as a subalgebra of FÐ\) where \ is a suitable Banach space.
2.2 Theorem: (Embedding an Banach Algebra in B(X,X)) Let E\E be a Banach algebra. Then there exists a Banach space such that is somorphic as a closed subalgebra of F\ß\ . ~ Proof - Let \œE (or let \œE if E does not have an identity. So assume W.L.O.G. that E has an identity). Consider the left-multiplication operator
XÀEÄEB CÈBC .
Clearly XC œ BCŸ B C llllllllB Therefore XXŸB is bounded and . BBll ll
10 Define 9 ÀEÄ F \ß\ by BÈXB 0) 9 is a homomorphism. This follows from the following observations:
XœXXXœXBC B C, α Bα B , and XœXXBC B C
1) 9 is injective.
XœXÊBDœCDaD−\BC ÊB/œC/ ÊBœC
2) 9" is continuous.
9ÐBÑ œ XB œsup XB C œ sup BC llllllCŸ" CŸ" ll ll ll
/" B ll B†// œ B†/œ / ½½ll llll ll
" ¾ÐBÑ B9 / llll ll Therefore 9" is bounded.
3) 9 E is complete. Since F\ß\ is complete, we need only show that a convergent sequence in 999EEE has its limit in . (i.e. show is closed). Suppose XÄX , then XCDœXCDaCßD−E . BBB888 Taking the limit, set XCDœXCD . Set Cœ/, then X D œX /D aD Let BœX/ , then XœXÞ 9B 9 4) 9 is bounded and surjective. Since 99" ÀEÄE is bounded and surjective, and 9 E and E are " Banach space, then 99" œ is continuous by the Open Mapping Theorem.
2.3 Corollary: In every Banach algebra with identity, there exists an equivalent norm for which /œ" . ll Proof - Define BœX . llN lB l Now, /œXÞ llN l/ l But, XB œ /B œ B aBÞ llllll/ Hence, Xœ"Þ ll/
11 Definition: B−E is invertible if there exists C−E such that BCœ/ and CBœ/ . In this case CB is denoted by " . ∞ For the next theorem we need to recall that a series B8 in a Banach spaces is 8œ" ∞ said to converge absolutely if B∞8 . A series that converges absolutely 8œ" ll converges (in the norm of the Banach space).
2.4 Theorem: Let EB−EB/"B be a Banach algebra and . If , then is ∞ ll invertible. And Bœ" /B 8 8œ! Proof - Let Cœ/B , then C œαα " ,. for some . 8 ll So, CŸCœÞ88α ll∞ ll Thus Dœ C8 converges absolutely in E . (Note that C! œ/ .) 8œ!
Now, BDœ /C Dœ/DCD ∞∞ œ/ C8 C C8 8œ! 8œ! ∞∞ œCC88(continuity of multiplication) 8œ! 8œ" œ/ Similarly, DB œ / . So D is the inverse of B .
Examples 1. EœF \ XM "ÊX is invertible. ll 2. If EœG!ß" , then cd 0" "Ê0 is invertible. ll Definition: The set K of invertible elements is called the group of invertible elements.
2.5 Corollary: The set KE of invertible elements of is open.
Proof - Let B−K . Consider the map XB ÀEÄ E where CÈBC . a) XB is continuous. Moreover,
XœBCŸBCC llllllllB Therefore XŸB and taking Cœ/ . we see that XœBÞ lB l ll lB l ll
b) XEB is onto . Proof - Let D−E , then X B" D œD B ¾XB is onto E . c) X is 1-1 it is in fact an isometry B
12 ¾X by the Open Mapping Theorem, B is open. i.e. X maps open sets to open sets Now Yœ B B/ " is an open neighborhood of I contained in K . ˜™¸ ll Thus XY is an open neighborhood of B . B Moreover Y§KÊBY§BKœK because B−K 2.6 Theorem: The map BÈB" defined on K is continuous.
Proof - Let B−K88 , where BÄC . " " We need to show BÄC8 First, suppose BÄ/Þ8
Then bR ! such that 8 R Ê / B " ll8 #
∞∞∞5 " 55" Then B8 œ /B88 Ÿ /B Ÿ# œ# ll¾¾ 5œ! 5œ! l l 5œ!ˆ‰
Therefore B" / œ B " /B Ÿ B " /B Ÿ# /B l88 ll 888 lllllll 8 Ä! as 8Ä∞Þ
" Thus, we have shown if BÄ/8 , then B8 Ä/Þ
" " Now suppose BÄC−K888 . Then C B−K and C BÄ/Þ "" " So, by the first part of the proof, C B8 Ä/ or B8 CÄ/ . " " " Multiplying by CBÄCÞ we get 8
Definition: A division algebra is an algebra where every nonzero element is invertible.
Example. 1. GÒ!ß "Ó is not a division algebra 2. ‚ is a division algebra.
2.7 Theorem (Gelfand-Mazur): Let EE be a complex Banach division algebra, then is isomorphic to ‚ . Proof - We show that every B−E is of the form Bœ--‚ / for some − . Suppose not, then bB − E such that B Á--‚ / a − Þ Hence, B--‚ /Á!a − Þ Let 0−Ew (the dual of E ) such that 0B" Á! . " Define 9‚: ‚ by 9-œ0 B - / ˆ‰ c d "Ñ Show 9‚ is entire (analytic on all of ).
13 " " 9-2 9- 0B2/-- 0B/ ‘ ‘ 22œ B-- 2 /" B / " œ0 ’“2 B-- / B 2 / " " œ0cd B-- / B 2 / ’“2 œ02/ B- / " B- 2 / " ‘2 œ0 B-- /" B 2 / " ‘ By Theorem 2.6, the limit exists as 2Ä! . namely as 2Ä!, B-- 2/" Ä B / " ‘ ¾ 9- is entire because this limit exists for all .
#Ñ Show that 9 is bounded.
" 9-œ0 B - /" œ"B 0 / ˆ‰ˆ‰ --’“
" " So, lim9- œ0/œ†0/ lim"B lim " lim B --Ä∞ Ä∞-- -- Ä∞ - Ä∞ - kk kk’“ˆ‰ kk kk ’“ ˆ‰ " œ†0/0−Elim"B lim since w - Ä∞ --- Ä∞ˆ‰ kk ’“kk " œ†0lim " lim B / by Theorem 2.6 - Ä∞ - - Ä∞ - kk ”•Škk ‹ œ!†0 / œ!Þ $Ñ By Louiville's Theorem, 99 is constant and so, by the above, ´ ! . This contradicts 0B" Á!Þ " note, by the definition of 99 , ! œ0 B!†/ œ0 B" œ! ] ‘ Exercises
1. The operator XB defined in Theorem 2.2 is linear.
2. The map 9 defined in Theorem 2.2 is a homomorphism.
3. GÒ!ß "Ó is not a division algebra. Which elements are invertible?
4. If llB-- then /B in invertible and ll ∞ Ð/BÑœ--" 8 B 8" 8œ"
5. If X−FÐ\Ñ where \ is a Banach space and if l-- l X then MX is invertible. ll " 6. If BCBC is invertible and B" then is invertible. llll
14 7. If E is a complex Banach algebra and if there exists -! such that BC - B C ll llll for all BC−E , , then E is isomorphic to ‚ .
8. An element B−E is a topological divisor of zero if there exists a sequence ÐBÑ8 with Bœ"BBÄ!BBÄ!Ñ888 and (or . (a) A topological divisor of zero is not invertible. (b)ll The boundary of the set of noninvertible elements is a subset of the set of topological divisors of zero.
15 3. Ideals
In this section we study special subalgebras called ideals. The concept of an ideal is very natural because, as we will see, the kernel of a homomorphism is always an ideal.
Definition: An ideal is an algebra EME is a subset of such that:
"Ñ M is a subalgebra of E #Ñ aB − Eß BM § M and MB § M
An ideal is proper if MÁ ! and MÁE . The term ideal (without qualification) will mean proper ideal. A properef ideal consists of noninvertible elements only, so a division algebra has no proper ideals.
Note that the concept of ideal is purely algebraic (i.e., does not involve the norm).
Definition: An ideal is maximal if it is not properly contained in any proper ideal.
Example. Ideals in GÒ!ß "Ó Let E œ G !ß " and let I § !ß " cd cd Let Mœ00Iœ! I ˜™¸ Let Qœ 00:œ!where :−!ß" : ˜™¸ c d M is an ideal because: If 1−Eß0−Mß then 01: œ0 :1: œ! . ¾01−M. I :: Note that if J§I , then MJI ¨M .
M: is a maximal ideal.
Proof - Suppose QQ§QQ:: is not maximal (i.e. for some ideal ). Let 0−QÏQ . Then 0: Á!Þ : Choose 1−Q such that 1B œ! , but 1> Á!a>Á: . : Clearly 0ß1 − QÞ Thus, 0## 1 − Q since is an algebra. But then 01>Á!a>−!ß"Ê01## ## is invertible. cd Thus, QœE .
If Q is maximal in EœG!ß" , then QœQ for some : . cd : Proof - For each 0−Qß consider the zero set of 0 : ^0 œ >0> œ! œ0" ! Þ ˜™¸ cef d Since 0! is continuous and is a closed set, then ^0 is a closed set. ef If 0ß1 − Qß then ^ 0 ∩^ 1 Á gÞ If not, 01>Á!a># ### . But then 01 is invertible, which is a contradiction. Thus the collection ^0 0−Q of closed sets has the finite intersection ˜™ ¸ property. Since !ß " is compact, then ^ 0 Á gÞ cd0−Q+
If :−O, then Q: ©QÞ
16 Since QOœ:ÞQœQÞ is maximal, then So, ef : 3.1 Theorem: The intersection of any collection of ideals is an ideal. Proof - Let MÀαA − be a collection of ideals. efα
Let Mœ MÞα αA+−
Suppose Bß C − Mß then Bß C − Mα aα Þ
But Mα is an ideal, so B C , BC − Mα aα Ê B Cß BC − M . Let D−E . Since B−MÊB−Mα aαα ÊBD−Mα a ÊBD−MÞ
3.2 Theorem: The closure of an ideal is an ideal. Proof - Let MBßC−M be an ideal and .
Then bB ß C −M such that B ÄBßC ÄC . 88 8 8 By continuity of addition and multiplication,
BCÄBC88 and BCÄBCÞ 88
Thus, BC−M and BC−MÞ
Suppose D−E , then DB8 −M and DB8 ÄDBÞ Thus DB − M.
3.3 Theorem: The closure of a proper ideal is a proper ideal. Proof - Let Yœ B B/ " consist of invertible elements and is open ˜™¸ll (Theorem 2.4). So EÏY is closed. So, for any proper ideal MM§EÏY ,
ÊM§EÏYœEÏY ÊM is proper .
3.4 Theorem: Every maximal ideal is closed. Proof - Let Q be a maximal ideal. By Theorem 3.3, Q is a proper ideal. But Q¨Q and Q is maximal. So, QœQÞ Thus, Q is closed.
3.5 Theorem: Every proper ideal is contained in a maximal ideal. Proof - Let M be an ideal. Consider the collection of all ideals containing M . This is a partially ordered set under inclusion. Apply Zorn's Lemma.
17 3.6 Theorem: Every noninvertible element B is contained in some maximal ideal. Proof - Let MœBEœ B++−E Þ ˜™¸ Then M is a proper ideal. By Theorem 3.5, MQB−M§QÞ is contained in a maximal ideal and
Exercises 1. If B is a noninvertible element in E then M œ BE œ ÖB+ À + − E× is a proper ideal.
2. Let E œ GÒ!ß "Ó and let B!! be the element in E defined by B Ð>Ñ œ >Þ Find the closure of the {BE! } generated by . 3. If I § J are subsets of Ò!ß "Ó then M ¨ M in GÒ!ß "Ó . µµIJ 4. If EœE"# ŠE then E " œÖÐBß!ÑÀB−E× " is an ideal in E . Also E " is isometrically isomorphic to E" . 5. Prove Theorem 3.5.
18 4. Quotient Algebras
First we review some basic facts about rings and then apply those results to Banach algebras. Note that an algebra is a ring with some additional properties.
Results from Ring Theory
Definition: Let EMEEÎM be a commutative ring and an ideal in . The quotient ring is defined as follows: EÎM œ B M B − E ˜™¸ With the following operations:
BM CM œ BC M BM CM œBCM ααBM œ BM With these operations, EÎM is a ring.
Note: BC and are in the same coset of MBC−MÞ iff
4.1 Theorem: If EF and are rings with identity and 9 ÀEÄF is a ring homomorphism then (1) the kernel of 9 is an ideal in E (2) If 99 is an epimorphism then EÎ5/<Ð Ñ is isomorphic to FÞ
Proof - (a) Let Bß C − 5/<Ð9 Ñ œ O , and D − E , then
a. 999BCœ B Cœ!!œ!¾BC−OÞ . b. 999BC œ B C œ ! † ! œ ! . ¾ BC − O c. 9999DB œ D B œ D † ! œ ! . ¾ DB − O ¾O is an ideal. . (b) 9 can be factored as follows:
1 EÄ EÎ5/<0 Ä2 F BÈBOÈ0 B 4.2 Theorem: QEEÎQ is a maximal ideal in the ring if and only if is a field.
19 Proof - (ÊBÂQÞ ) Suppose Let [ be the ideal generated by B and Q .i.e. [ œ CBDC−QßD−E ˜™¸
Now, setting Cœ! and Dœ/ we see that !B/œB−[ , so [ properly contains QQ . Since is maximal, [œEÞ
So, bC99 − M such that / œ C9 D BÞ
Ê/Q œC9 DBQ9
œC99 QDBQ
œQDBQ9
œDBQ9
œDQBQ 9
Note that / Q is the multiplicative identity of EÎQ . So, D9 Q is the inverse of BQÞ Ð É Ñ If O is an ideal properly containing Q , then OÎQ is an ideal in EÎQ . But EÎQ is a field so it has no proper ideals. This contradiction proves that OQ doesn't exist, so is maximal.
Quotients of Banach Algebras
Recall from Section 1 that if EME is a Banach algebra and an ideal in , then the quotient algebra EÎM is a normed algebra with the norm
BM œinf BC llllC−M
4.3 Theorem: If MEÎM is a closed ideal, then is a Banach algebra. Proof - Exercise.
4.4 Theorem: If Q is a maximal ideal, then EÎQ ¸ ‚ . Proof - Note that EÎQ is a Banach algebra and EÎQ is a field. By the Gelfand-Mazur Theorem, EÎQ ¸ ‚ .
Exercises
1. If MEBCM is an ideal in the ring then and are in the same coset of if and only if BC−M
2. What is GÒ!ß "ÓÎM!! where M œ ÖB − GÒ!ß "Ó À BÐ!Ñ œ !× .
20 " 3. Let I œ Ò!ß# Ó . What is GÒ!ß "ÓÎMII where M œ ÖB − GÒ!ß "Ó À BÐIÑ œ !× , and .
4. Prove Theorem 4.3
21 5. Multiplicative Linear Functionals
Linear functionals on a Banach algebra which preserve the multiplicative structure of the algebra play a special role. We will see that they are in one-to-one correspondence with the maxiaml ideals of the algebra.
Definition: LetE be a complex commutative Banach algebra with identity. A nonzero linear functional 0ÀEÄ‚ which satisfies 0BCœ0B0C aBßC−E is called a multiplicative linear functional. (i.e. 0E is a homomorphism from into ‚ ).
5.1 Theorem: The kernel of a nonzero m.l.f. is a maximal ideal. Proof - Let 0ÀEÄ‚ be a m.l.f. By Theorem 4.1(1), 5/<Ð0Ñ is an ideal. By Theorem 4.1(2),
EÎ5/< 0 ¶ ‚ . Since ‚ is a field, 5/< 0 is a maximal ideal (by Theorem 4.2). Note: Recall from Functional Analysis that if 00À\Ä is a linear functional ‚ and if 5/< 0 is closed, then 0 is continuous. We use this fact in the next theorem. 5.2 Theorem: Every nonzero multiplicative linear functional is continuous. Proof - Let 0ÀEÄ‚ be a multiplicative linear functional. Since 5/< 0 is a maximal ideal, then it is closed (Theorem 3.4). It follows from the above theorem that 0 is continuous.
5.3 Theorem: If 00œ"Þ is a multiplicative linear functional, then ll Proof - 1) Show that 0Ÿ" . i.e. 0BŸBaB−E . ll k k ll For suppose not, then bB such that 0 B B 99kkll 9 Let α‚−0B"B such that α α kk 99 ll Ê0αα B88 œ0 B Ä∞ kkkkcd 9 9 88 Whereas ααBŸBÄ!99 (by submultiplicativity of the norm) This contradictsllll the continuity of the norm. ¾0Ÿ"Þ ll 2) Now, 0/ œ" and / œ"Þ ll So, 0/ œ /ß then 0 "Þ kkllll ¾0œ"Þ ll
22 Note:
Given a multiplicative linear functional 0ÀEÄ‚ , we know that its kernel Q0 is maximal. Given a maximal ideal Q§E , we define
0ÀEÄEÎQ¶Q ‚ where
BÈBQ
So 0EÞQ is a multiplicative linear functional on
This correspondence between multiplicative linear functionals and maximal ideals in injective, i.e.
0œ0QœQQ0 and 0Q .
We have shown that there is a 1-1 correspondence between the multiplicative linear functionals on EE and the maximal ideals of .
Exercises
1. If 00/œ" is a m.l.f. then 2. If 0B is a m.l.f. and is invertible, then 0Bœ0B" " c d 3. If QE is a maximal ideal in , explain how the quotient map 0ÀEÄEÎQ is a m.l.f.
4. A commutative Banach algebra must have at least one m.l.f. defined on itÞ
5. For >!> − Ò!ß "Ó the evaluation functional 0! À GÒ!ß "Ó Ä ‚ is defined by
0ÐBÑœ BÐ>! Ñ . The evaluation functional is a m.l.f.. What is its kernel?
6. Every m.l.f. on GÒ!ß "Ó is an evaluation functional.
7. Every m.l.f. on PÐ"3:>‘‘ Ñ is given by 0ÐBÑœ∞ BÐ>Ñ/ .> for some :− . '∞ 8. There are no m.l.f.'s on P" Ð!ß "Ñ . (Note that this algebra has no identity.)
∞ "38> 9. Every m.l.f. in 6Й1 Ñ has the form 0ÐBÑœ B/8 for some >ß!Ÿ>#Þ 8œ∞
10. Every m.l.f. on the algebra EÐHÑ is of the form 0ÐBÑ œ BÐ:Ñ for some : − H .
23 6. Banach's Theorem on Semisimple Banach Algebras
Let Eß F Banach algebras. When is a homomorphism9 À E Ä F continuous? We have already seen that if Fœ‚9 , then , which is a multiplicative linear functional, is automatically continuous. In this section we see what exactly is special about ‚ that make this property of automatic continuity possible.
The Separating Space
To study continuity of homomorphisms, we consider the places they are not continuous.
Definition: Let XÀ\Ä] be a linear transformation between the Banach spaces \ and ]X . The separating space of is given by:
Æ Xœ C−]bB §\ßBÄ!and XBÄCÞ ˜™¸ 88 8 6.1 Theorem: Let XÀ\Ä], then 1) Æ X] is a closed linear subspace of . 2) XXœ!Þ is continuous iff Æ ef Proof - 1) Let ÐC Ñ be a sequence in Æ X such that C Ä C . 88 Since each CX is in Æ then for each 8B−\ we find such that 88 B"" and XÐBÑC. ll88888 l l Clearly BÄ! and XÐBÑÄC . Hence C−ÆÆ X . So X is closed. 88 2) If XXœ! is continuous, then Æ . ef Suppose C−Æ X ß then bB §\ such that B Ä ! and T B Ä C . 888 Since XXBÄX!œ!Cœ!Þ is continuous, . So, 8 Conversely, if Æ Xœ! , then X is continuous by the Closed Graph Theorem. 6.2 Theorem: Let XU be a linear transformation and the quotient map as follows:
U \Ä]X Ä]Î[ , where []UXX§[ is a closed subspace of . If is continuous, then Æ . Proof - Since UX is continuous, Æ UX œ ! [ . e f If BÄ! in \ , then UXB Ä![ 88
If XB8 Ä C , the C − [ (because UÐCÑœ ![ )
¾X§[ÞÆ
24 6.3 Theorem: Let XW and be mappings such that
\Ä]X Ä^W , where WWXX§5/ \Ä]X Ä]Î5/ Semisimple Banach Algebras Definition: A Banach algebra E is called semisimple iff Q§EQ is a maximal ideal œ !Þ +˜™¸ ef note: VE œ Q is called the Jacobson Radical of E . + Example G !ß" is semisimple because if 0 −Q for each :− !ß" , then 0 : œ!a:− !ß" cd: cd cd which implies 0œ ! . ef 6.4 Theorem: If 9 is a homomorphism from the Banach algebra E to the semisimple Banach algebra F , then 9 is continuous. Proof - Consider a multiplicative linear functional 0F on , and consider 9 0 EÄ FÄ ‚. Now 0‰99 is a multiplicative linear functional on E , so 0‰ is continuous. By Theorem 6.3, Æ9 §5/<0Þ But 5/<0 œ Q is a maximal ideal. Moreover, every maximal ideal is the kernel of some multiplicative linear functional. In other words, Æ9 is contained in every maximal ideal of F§∩Qœ! . Thus, Æ9 . So, 9 is Q maximal ef continuous. The theorem just proved illustrates the power of the functional analysis approach. We were able to prove the continuity of a whole class of operators by cosidering the Banach algebra structure only, and not individual elements in the algebra. (i.e., we didn't need any %$ - arguments.) Exercises 1. If XÀ\Ä], then Æ X] is a closed linear subspace of . 2. If 99ÀEÄF is an algebra isomorphism and if E (or F ) is semisimple then is cont. 3. If EÀEÄE is a semisimple Banach algebra then any homomorphism 9 is cont. 4. The Banach algebra GÒ!ß "Ó is semisimple. 5. The Banach algebra EÐHÑ is semisimple. 25 7. The Gelfand Theory The goal is to represent any Banach algebra E as a subalgebra of the Banach algebra G\ where \ is a compact Hausdorff space. Recall that G\ was given as an example of a Banach algebra. We want to show that any Banach algebra is a subalgebra ofG \ . Thus, we need to find \ and a homomorphism E Ä GÐ\ÑÞ Recall, we defined the maximal ideal space of E as: QE œ QQ is a maximal ideal in E ˜™¸ œ00 is multiplicative linear functional on E ˜™¸ It is possible to equip QE with a topology so that it becomes a compact topological space. The topology is the AAE‡‡w -topology inherited from the -topology on the dual of EQE§EÞA. Note that w‡ The -topology is described in Appendix 1. w Note that if Ð088 Ñ is a sequence in E then 0 Ä 0 in the A *-topology iff 0 8 ÐBÑ Ä 0ÐBÑ for each B − \ . (This is because each B − \ represents a functional on Ew in N ÐEÑÞÑ 7.1 Theorem: Let EQEA be a Banach algebra. Then with the *-topology is a compact Hausdorff space. Proof - By the Banach-Alaogolu theorem the unit ball YE of w is A *-compact. We show that QÐEÑ is a closed subset of the Y . Cleary QÐEÑ § Y because for every multiplicative linear functional 0Ÿ" . Now suppose Ð0 Ñ is a sequence in QÐEÑ and ll 8 0Ä08 . We need to show that 0−QÐEÑ ; so we need to show that 0 is a multiplicative linear functional. We show that 00 is multiplicative and leave the other ( is linear as an exercise). For any Bß C − \ we have 088 ÐBÑ Ä 0ÐBÑ , 0 ÐCÑ Ä 0ÐCÑ so 088 ÐBÑ0 ÐCÑ Ä 0ÐBÑ0ÐCÑ We also have 08888 ÐBCÑ Ä 0ÐBCÑ and since 0 ÐBCÑ œ 0 ÐBÑ0 ÐCÑ so we must have 0ÐBÑ0ÐCÑÄ0ÐBCÑ88 By the uniquness of limits we must have 0ÐBCÑ œ 0ÐBÑ0ÐCÑ . Thus 0 is multiplicative. In the same way we show that 0 is linear. Thus 0 − QÐEÑ so QÐEÑ is closed in Y . Finally, since EAw is Hausdorff in the *-topology, it follows that QÐEÑ is also Hausdorff. 26 Definition: Let B−E . Define the Gelfand transform s B of B to be the function BÀQsss E Ä‚ where 0ÈB0 , where B0 œ0 BÞ 7.2 Theorem: Each BQEB−GQEÞss is a continuous function on . So, Proof - If 088 Ä 0 in QÐEÑ then 0 ÐBÑ Ä 0ÐBÑ for each B − \ . But this last statement means that B0ss ÄBÐ0Ñ . So Bs is continuous. 8 Example Let EœG!ß". cd Q E œ 0 : − !ß " where 0 B œ B : Þ ˜™::¸ c d So we can identify Q E with !ß " as : È 0 . With the A‡ -topology, Q E is actually c d: homeomorphic to !ß " Þ This shows we are able to recover the space !ß " from only the cd cd algebraic structure of E . So, we can replicate this procedure with any Banach algebra. i.e. start with the maximal ideals, impose a topology on QE , etc... Now, BÀQsss E Ä‚ , where 0ÈB0 œ0 BÞ In particular, B0 œ0 B œB:Þ :: i.e. BÀ:ÈB:ss , where B: œB: . So that BœBs . Thus, we are able to recover G!ß" from the algebraic structure of E . cd It can be observed that BQEÞs is a bounded function on Proof - B0s œ 0B Ÿ B since 0 is continuous and 0 œ"Þ kkkkll ll Thus, B0s is bounded by BÞ ll 7.3 Theorem: The map > ÀEÄ G Q E where BÈBs is a continuous algebra homomorphism from the Banach algebra EGQE into the Banach algebra with the supnorm. i.e. Bœss sup B0Þ Also, if E is semi-simple, then > is injective. ll0−Q E k k Proof - 1) Show > is a continuous algebra homomorphism. BCs 0 œ0 BC œ0 B 0 C œB0ss C 0 BCs 0 œ0BCœ0B0C œB0ss C 0 αααB0œ0Bœ0Bs œB0αs Bœs supB0s œ sup 0B ll 0−Q E kk 0−Q E kk ŸB since 0BŸBa0Þ i.e. all m.l.f. have norm " . ll k k ll 2) Show if E is semi-simple, then > is injective. Suppose B´!^^ . i.e. Bœ!a0−Q E Ê0 B œ!a0−Q E ÊB−5/<0 a0−Q E ÊB−Q a maximal ideals of Q ÊB−V+.E ÊBœ! 27 7.4 Theorem: BB0Á!a0−QE is invertible iff s . Proof - ÊB0œ!0−QE Suppose s for some Then 0 B œ!ÊB−5/<0 , which is a maximal ideal ÊB is not invertible ÉB Suppose is not invertible. Then B− some maximal ideal Q (By Theorem 3.6). But then B−5/<0. i.e. 0 B œ!Þ QQ Exercises 1. Let \] and be compact Hausdorff spaces. If GÐ\Ñ is algebraically isomorphic to GÐ]Ñ then \ and ] are homeomorphic. 2. \GÐ\Ñ is is a connected topological space iff has no zero divisors. 3. Banach algebraEQÐEÑ has no zero divisors if and only if is connected. 4. If the Gelfand representation of a semisimple Banach algebraE is complete in the spectral norm, then the original norm of E is equivalent to the spectral norm. 5. If 7 is any comact Hausdorff topology on QÐEÑ for all the Gelfand transforms s B are continuous then 7 coincides with the Gelfand topology on QÐEÑ . 6. If \GÐ\Ñ\ is compact then the maximal ideal space of is homeomorphic with . [Hint: Use Exercise 5]. 7. If EE has no identity and " is the algebra obtained by adjoining an identity then QÐE" Ñ is the one-point compactification of QÐEÑ . 8. The Gelfand transform is an isometry if and only if BœB# # for every B−EÞ ll ll 9. The following are equivalent: (1) > is 1-1, (2) QÐEÑ separates the point of E , (3) <ÐBÑ is a norm on E . 10. The 25 -topology is weaker than the A‡ -topology, so the two topologies coincide if the 25 -topology is Hausdorff. 28 8. Wiener's Theorem on Absolutely Convergent Trigonometric Series Let [ be the collection of all absolutely convergent trigonometric series, that is, series of the form ∞ 38> B> œ +/8 >−!ß#Ñ1 8œ∞ c ∞ with B œ +8 ∞Þ With this norm [ is a Banach space. It becomes an algebra if ll 8œ" k k we define ∞∞ 38> 38> B>C> œ +/88 ,/ Œ Œ 8œ∞ 8œ∞ ∞∞ 38> œ+,/85 5 (Cauchy product) 8œ∞ Œ 5œ∞ This norm is an algebra norm: 1) Bœ! iff Bœ!Þ ll 2) ααBœ B ll kkll ∞ 3) BC œ +88 , ll8œ∞ k k ∞∞ Ÿ+,œBC88 8œ∞ kk 8œ∞ kk llll ∞∞ 4) BC œ +85 , 5 ll 8œ∞ ºº5œ∞ ∞∞ Ÿ+,85 5 8œ∞ 5œ∞kkkk ∞∞ œ+,85 5 5œ∞ 8œ∞kkkk ∞∞ œ,585 + 5œ∞ kk8œ∞ k k ∞∞ œ,585 + 5œ∞ kkŒ 8œ∞ k k œB C llll ¾[ is a Banach algebra with this norm called the Wiener algebra. 29 8.1 Theorem: Every multiplicative linear functional on [ is of the form 0BœB> >99 where >−!ß#Ñ1 . 9 c Proof - Ê0 We first show that is indeed a multiplicative linear functional on [ . >! 0 BC œ BC > œB > C > œ0 B 0 C >99>>9 9 99 0BœB>œ0Bαα α >9>99 0 BC œBC> œB> C> œ0 B0 C >99>>9 9 99 Next we show that every multiplicative linear functional is of this form. Let 0[ be a m.l.f. on . Let B > œ/3> −[ and let # œ0 B Þ 99 Then # œ0B Ÿ B œ" kk k 99 k l l #"œ0B" Ÿ B " œ" kkk ! kll9 ¾œ"# kk Since #‚−œ/>−!ß#Ñ , then #3>9 for some 1 9 c Thus, 0B œ/3>9 ! 8 8 0B œ0/3> œ/ 38>9 ! RRRR 8 8 38> 3> 38>9 0+Bœ0+/œ+0/œ+/8888! Œ Œ 8œR 8œR 8œR 8œR ∞ 38> By continuity of 0B>œ+/ , if we let 8 8œ∞ ∞ 38>9 0B œ +/89 œB> 8œ∞ i.e. 0œ0>9 8.2 Theorem: (Wiener's Theorem) Let B−[ , if B> Á!a>−Ò!ß#1 Ñß then " B> −[. i.e. if B is an absolutely convergent trigonometric series that never vanishes, then its reciprocal is also an absolutely trigonometric series. Proof - By the Gelfand Theorem, suppose B> Á!a>−Ò!ß#Ñ1 Consider B0−Q[0Þs . Since each is of the form >9 Thus, B0s œ0 B œB> >99 >9 So, if BB never vanishes, s never vanishes. So by Theorem 7.4, B is invertible in [ . Exercises 1. Show how the Cauchy product is obtained by directly multiplying the two series and collecting like terms. 30 9. Finitely-Generated Banach Algebras Let EO§E be a Banach algebra. A subset is called a set of generators for a subalgebra EE!! if is the smallest subalgebra of E containing O and / . We write EœO! . If EœO we say that E is generated by O . If E is generated by OœÖB×, a single element, we way that E is singly generated . Theorem 9.1: If EÖBßáßB×QÐEÑ is finitely-generated by "8 then is is homeomorphic to a compact subset of ‚8 . Proof - The map 9‚ÀQÐEÑÄ 8 0 È ÐBss"# Ð0Ñß B Ð0Ñß á ß sB 8 Ð0ÑÑ is continuous because it is a map into a product space and the composition with each projection is continuous (Appendix 1, Section 4, Exercise 2). It follows that 9 is a homeomorphism onto its image by (Appendix 1, Section 5, Exercise 2). Exercises 1. EO is generated by if and only if the set of all finite sums of the form + BBâB55"# 58 5ß5ßá5"# 8 "# 8 is dense in E . Here +5ß5ßá5"# 8 −‚ and B " ßB # ßáßB 8 −O . 2. The Banach algebra GÒ!ß "Ó is singly-generated. 3. The Banach algebra GГ Ñ is not singly-generated. 31