Geometric Properties of the Disk Algebra

Geometric Properties of the Disk Algebra.

Yun Sung Choi (Joint Work with D. Garcia, S.K. Kim and M. Maestre)

POSTECH

Worksop on Functional Analysis Valencia June 3-6, 2013

Worksop on Functional Analysis Valencia June 3-6, 2013 1 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20

Related to the numerical index, Daugavet property, Bishop-Phelps-Bollob´as property, etc. several geometrical properties of the space C(K)(resp.C(K, X )) on a Hausdor↵ compact set K (resp. with values in a Banach space X )have been obtained, Most of the results at one point or another use the classical Urysohn Lemma.

Urysohn Lemma

Theorem (Urysohn Lemma) AHausdor↵ topological space is normal if and only if given two disjoint closed subsets can be separated by a continuous function with values on [0, 1] and taking the value 1 on one closed set an 0 on the other.

Worksop on Functional Analysis Valencia June 3-6, 2013 2 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Urysohn Lemma

Worksop on Functional Analysis Valencia June 3-6, 2013 2 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 For example consider the most representative case, the disk algebra A(D) of functions continuous on the closed unit complex disk D and holomorphic in the open disk D of C.

Why Complex Version ? (Stone-Weierstrass Theorem) The only uniform algebra in the real case is C(K)

Why a complex version of Urysohn Lemma

Many of that results could not be extended to a uniform algebra (i,e. to a closed subalgebra of a complex C(K) that separates points and contains the constant function 1) since the Urysohn Lemma cannot be true, in general, if we ask the function to be in a given uniform algebra.

Worksop on Functional Analysis Valencia June 3-6, 2013 3 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Why Complex Version ? (Stone-Weierstrass Theorem) The only uniform algebra in the real case is C(K)

Why a complex version of Urysohn Lemma

For example consider the most representative case, the disk algebra A(D) of functions continuous on the closed unit complex disk D and holomorphic in the open disk D of C.

Why Complex Version ? (Stone-Weierstrass Theorem) The only uniform algebra in the real case is C(K)

Worksop on Functional Analysis Valencia June 3-6, 2013 3 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 A uniform algebra is a closed subalgebra A C(K)thatseparatesthe ⇢ points of K and contains the constant function 1.

Given x K, we denote by x : A C the evaluation functional at x 2 ! given by (f )=f (x), for f A. x 2 The natural injection i : K A deﬁned by i(t)= for t K is a ! ⇤ t 2 homeomorphism from K onto (i(K), w ⇤).

Complex Version of Urysohn Lemma

Here, C(K) stands for the space of complex-valued continuous functions deﬁned on a compact Hausdor↵ space K equipped with the supremum norm . k · k1

Worksop on Functional Analysis Valencia June 3-6, 2013 4 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Given x K, we denote by x : A C the evaluation functional at x 2 ! given by (f )=f (x), for f A. x 2 The natural injection i : K A deﬁned by i(t)= for t K is a ! ⇤ t 2 homeomorphism from K onto (i(K), w ⇤).

Complex Version of Urysohn Lemma

Here, C(K) stands for the space of complex-valued continuous functions deﬁned on a compact Hausdor↵ space K equipped with the supremum norm . k · k1 A uniform algebra is a closed subalgebra A C(K)thatseparatesthe ⇢ points of K and contains the constant function 1.

Worksop on Functional Analysis Valencia June 3-6, 2013 4 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 The natural injection i : K A deﬁned by i(t)= for t K is a ! ⇤ t 2 homeomorphism from K onto (i(K), w ⇤).

Complex Version of Urysohn Lemma

Given x K, we denote by x : A C the evaluation functional at x 2 ! given by (f )=f (x), for f A. x 2

S = x⇤ A⇤ : x⇤ =1, x⇤(1)=1 , { 2 k k } then the set (A) of all t K such that is an extreme point of S is a 0 2 t boundary for A that is called the Choquet boundary of A.

Complex Version of Urysohn Lemma

AsetS K is said to be a boundary for the uniform algebra A if for every ⇢ f A there exists x S such that f (x) = f . 2 2 | | k k1

Worksop on Functional Analysis Valencia June 3-6, 2013 5 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Complex Version of Urysohn Lemma

AsetS K is said to be a boundary for the uniform algebra A if for every ⇢ f A there exists x S such that f (x) = f . 2 2 | | k k1 For a uniform algebra A of C(K), if

S = x⇤ A⇤ : x⇤ =1, x⇤(1)=1 , { 2 k k } then the set (A) of all t K such that is an extreme point of S is a 0 2 t boundary for A that is called the Choquet boundary of A.

Theorem (Cascales, Guirao, and Kadets, 2012) Let A C(K) be a uniform algebra for some compact Hausdor↵ space K ⇢ and 0 = 0(A). Then, for every open set U KwithU = and 0 < ✏ < 1,there ⇢ \ 0 6 exist f Aandt0 U 0 such that f (t0)= f =1, f (t) < ✏ for 2 2 \ k k1 | | every t K Uand 2 \

f (K) R✏ = z C : Re(z) 1/2 +(1/p✏) Im(z) 1/2 . ⇢ { 2 | | | |  } In particular,

f (t) +(1 ✏) 1 f (t) 1, for all t K. | | | |  2

Worksop on Functional Analysis Valencia June 3-6, 2013 6 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 If A is a uniform algebra, then AX is deﬁned to be a subspace of C(K, X ) such that

X A = f C(K, X ):x⇤ f A for all x⇤ X ⇤ . { 2 2 2 }

Given f A and x X ,wedeﬁnef x C(K, X )by(f x)(t)=f (t)x 2 2 ⌦ 2 ⌦ for t K.Wewrite 2 A X = f x ; f A, x X . ⌦ { ⌦ 2 2 } From the deﬁnition of AX we note that A X AX . ⌦ ⇢

Complex Version of Urysohn Lemma - Uniform Algebra

We apply this Urysohn type Lemma to extend results on C(K)orC(K, X ) to AX concerning the numerical index, Daugavet equation, lushness and the approximate hyperplane series property (in short AHSP).

Worksop on Functional Analysis Valencia June 3-6, 2013 7 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Given f A and x X ,wedeﬁnef x C(K, X )by(f x)(t)=f (t)x 2 2 ⌦ 2 ⌦ for t K.Wewrite 2 A X = f x ; f A, x X . ⌦ { ⌦ 2 2 } From the deﬁnition of AX we note that A X AX . ⌦ ⇢

Complex Version of Urysohn Lemma - Uniform Algebra

We apply this Urysohn type Lemma to extend results on C(K)orC(K, X ) to AX concerning the numerical index, Daugavet equation, lushness and the approximate hyperplane series property (in short AHSP).

If A is a uniform algebra, then AX is deﬁned to be a subspace of C(K, X ) such that

X A = f C(K, X ):x⇤ f A for all x⇤ X ⇤ . { 2 2 2 }

We apply this Urysohn type Lemma to extend results on C(K)orC(K, X ) to AX concerning the numerical index, Daugavet equation, lushness and the approximate hyperplane series property (in short AHSP).

If A is a uniform algebra, then AX is deﬁned to be a subspace of C(K, X ) such that

X A = f C(K, X ):x⇤ f A for all x⇤ X ⇤ . { 2 2 2 }

Given f A and x X ,wedeﬁnef x C(K, X )by(f x)(t)=f (t)x 2 2 ⌦ 2 ⌦ for t K.Wewrite 2 A X = f x ; f A, x X . ⌦ { ⌦ 2 2 } From the deﬁnition of AX we note that A X AX . Worksop on Functional Analysis Valencia June 3-6, 2013 7 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the⌦ Disk (POSTECH)⇢ Algebra. /20 (1) Au(BX )hask-numerical index 1 for every k, has the lush property and has the AHSP property.

(2) The algebra of the disk A(D), and more in general any uniform algebra such that its Choquet boundary has no isolated points, has the polynomial Daugavet property.

Main Results

we study some geometrical properties of certain classes of uniform algebras, in particular the disk algebras Au(BX ) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space X .

Worksop on Functional Analysis Valencia June 3-6, 2013 8 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 (2) The algebra of the disk A(D), and more in general any uniform algebra such that its Choquet boundary has no isolated points, has the polynomial Daugavet property.

Main Results

(1) Au(BX )hask-numerical index 1 for every k, has the lush property and has the AHSP property.

Worksop on Functional Analysis Valencia June 3-6, 2013 8 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Main Results

(1) Au(BX )hask-numerical index 1 for every k, has the lush property and has the AHSP property.

(2) The algebra of the disk A(D), and more in general any uniform algebra such that its Choquet boundary has no isolated points, has the polynomial Daugavet property.

Worksop on Functional Analysis Valencia June 3-6, 2013 8 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Given a bounded function : S X ,itsnumerical range is deﬁned by X !

V ():= x⇤((x)) : (x, x⇤) ⇧(X ) { 2 } and its numerical radius is deﬁned by

v():=sup : V () . {| | 2 }

Let us comment that for a bounded function : ⌦ X ,where ! S ⌦ X , the above deﬁnitions are applied by just considering X ⇢ ⇢ V ():=V ( ). |SX

Numerical Index

For a Banach space X ,wewrite⇧(X ) to denote the subset of X X ⇥ ⇤ given by

⇧(X ):= (x, x⇤):x S , x⇤ S , x⇤(x)=1 . { 2 X 2 X ⇤ }

Worksop on Functional Analysis Valencia June 3-6, 2013 9 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Let us comment that for a bounded function : ⌦ X ,where ! S ⌦ X , the above deﬁnitions are applied by just considering X ⇢ ⇢ V ():=V ( ). |SX

Numerical Index

For a Banach space X ,wewrite⇧(X ) to denote the subset of X X ⇥ ⇤ given by

⇧(X ):= (x, x⇤):x S , x⇤ S , x⇤(x)=1 . { 2 X 2 X ⇤ }

Given a bounded function : S X ,itsnumerical range is deﬁned by X !

V ():= x⇤((x)) : (x, x⇤) ⇧(X ) { 2 } and its numerical radius is deﬁned by

v():=sup : V () . {| | 2 }

Worksop on Functional Analysis Valencia June 3-6, 2013 9 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Numerical Index

For a Banach space X ,wewrite⇧(X ) to denote the subset of X X ⇥ ⇤ given by

⇧(X ):= (x, x⇤):x S , x⇤ S , x⇤(x)=1 . { 2 X 2 X ⇤ }

Given a bounded function : S X ,itsnumerical range is deﬁned by X !

V ():= x⇤((x)) : (x, x⇤) ⇧(X ) { 2 } and its numerical radius is deﬁned by

v():=sup : V () . {| | 2 }

Let us comment that for a bounded function : ⌦ X ,where ! S ⌦ X , the above deﬁnitions are applied by just considering X ⇢ ⇢ V ():=V ( SX ). | Worksop on Functional Analysis Valencia June 3-6, 2013 9 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 If we consider elements of (B , X ) instead of continuous Au X k-homogeneous polynomials, we can deﬁne the analytic numerical index of X by n (X )=inf v(f ):f (B , X ), f =1 . a { 2 Au X k k }

Numerical Index

For k N,wedeﬁne 2 n(k)(X )=inf v(P):P (k X ; X ), P =1 , { 2 P k k } where (k X ; X ) is the space of all continuous k-homogeneous polynomials P from X into X ,andcallitthepolynomial numerical index of order k of X . When k = 1, it is the numerical index of X .

Worksop on Functional Analysis Valencia June 3-6, 2013 10 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Numerical Index

If we consider elements of (B , X ) instead of continuous Au X k-homogeneous polynomials, we can deﬁne the analytic numerical index of X by n (X )=inf v(f ):f (B , X ), f =1 . a { 2 Au X k k }

Worksop on Functional Analysis Valencia June 3-6, 2013 10 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 i.e. na(X )canbecalledthe“non-homogeneous polynomial numerical index of X ”.

Clearly, n (X ) n(k)(X ) a  for every k N. 2 We also denote by (X ) the space of scalar-valued polynomials on X . P

Numerical Index

Since the space (X ; X ) of all continuous polynomials from X into X is P dense in (B , X )wehavethat Au X n (X )=inf v(P):P (X ; X ), P =1 , a { 2 P k k }

Worksop on Functional Analysis Valencia June 3-6, 2013 11 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Numerical Index

Since the space (X ; X ) of all continuous polynomials from X into X is P dense in (B , X )wehavethat Au X n (X )=inf v(P):P (X ; X ), P =1 , a { 2 P k k }

i.e. na(X )canbecalledthe“non-homogeneous polynomial numerical index of X ”.

Clearly, n (X ) n(k)(X ) a  for every k N. 2 We also denote by (X ) the space of scalar-valued polynomials on X . P

Worksop on Functional Analysis Valencia June 3-6, 2013 11 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 (k) (k) X Sketch of Proof (n (X ) n (A )foreveryk 1) Let P S (k AX ,AX )  2 P✏ and 0 < ✏ < 1 be given. Choose f S X so that P(f ) > 1 .Find 0 2 A k 0 k 6 t such that P(f )(t ) > 1 ✏ .SinceP is continuous at f ,there 1 2 0 0 1 6 0 exists 0 < < 1 such that P(f ) P(g) < ✏ for every g AX with k 0 k 6 2 f g < . k 0 k Let

W = t K : f (t) f (t ) < /6, P(f )(t) P(f )(t ) < ✏/3 . { 2 k 0 0 1 k k 0 0 1 k } This set is open in K and t W . 1 2 \ 0

Numerical Index

Theorem Suppose that A be a uniform algebra. Then n(k)(AX ) n(k)(X ) for every k 1 and n (AX ) n (X ). a a

Worksop on Functional Analysis Valencia June 3-6, 2013 12 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Let

W = t K : f (t) f (t ) < /6, P(f )(t) P(f )(t ) < ✏/3 . { 2 k 0 0 1 k k 0 0 1 k } This set is open in K and t W . 1 2 \ 0

Numerical Index

Theorem Suppose that A be a uniform algebra. Then n(k)(AX ) n(k)(X ) for every k 1 and n (AX ) n (X ). a a (k) (k) X Sketch of Proof (n (X ) n (A )foreveryk 1) Let P S (k AX ,AX )  2 P✏ and 0 < ✏ < 1 be given. Choose f S X so that P(f ) > 1 .Find 0 2 A k 0 k 6 t such that P(f )(t ) > 1 ✏ .SinceP is continuous at f ,there 1 2 0 0 1 6 0 exists 0 < < 1 such that P(f ) P(g) < ✏ for every g AX with k 0 k 6 2 f g < . k 0 k

Worksop on Functional Analysis Valencia June 3-6, 2013 12 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Numerical Index

W = t K : f (t) f (t ) < /6, P(f )(t) P(f )(t ) < ✏/3 . { 2 k 0 0 1 k k 0 0 1 k } This set is open in K and t W . 1 2 \ 0

Worksop on Functional Analysis Valencia June 3-6, 2013 12 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Deﬁne (x)=x (x) 1 (1 )f + x AX for x X . 1⇤ 6 0 2 2 Deﬁne Q (k X ; X)by 2 P Q(x)=P (x) (t ), (x X ). (1) 0 2

Sketch of Proof

By the complex version of Urysohn Lemma there exist a function : K D and t0 W 0 ! 2 \ such that A, (t ) = 1, (w) < for every w K W ,and 2 0 | | 6 2 \ (t) + 1 1 (t) 1 | | 6 | |  for every t K. 2

Sketch of Proof

By the complex version of Urysohn Lemma there exist a function : K D and t0 W 0 ! 2 \ such that A, (t ) = 1, (w) < for every w K W ,and 2 0 | | 6 2 \ (t) + 1 1 (t) 1 | | 6 | |  for every t K. 2 Deﬁne (x)=x (x) 1 (1 )f + x AX for x X . 1⇤ 6 0 2 2

Worksop on Functional Analysis Valencia June 3-6, 2013 13 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Sketch of Proof

Worksop on Functional Analysis Valencia June 3-6, 2013 13 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Corollary For any Banach space X , and k 1, the following hold. 1 For a uniform algebra A we have n(k)(A)=1for every k 1 and na(A)=1. (k) n (k) n 2 n ( (D , X )) = n (X ) and na( (D , X )) = na(X ) for every A A n N. 2 3 n(k) (B ) =1for every k 1 and n (B ) =1. Au X a Au X

Numerical Index

Theorem Let A be a uniform algebra and X be a Banach space. Assume that AX has the following property: For every P (k X ; X ) and t K, 2 P 2 Q : AX C(K, X ) where Q(f )(t)=P(f (t)) satisﬁes that Q(f ) AX ! 2 for every f AX .Thenn(k)(AX )=n(k)(X ). 2

Worksop on Functional Analysis Valencia June 3-6, 2013 14 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Numerical Index

Corollary For any Banach space X , and k 1, the following hold. 1 For a uniform algebra A we have n(k)(A)=1for every k 1 and na(A)=1. (k) n (k) n 2 n ( (D , X )) = n (X ) and na( (D , X )) = na(X ) for every A A n N. 2 3 n(k) (B ) =1for every k 1 and n (B ) =1. Au X a Au X

Worksop on Functional Analysis Valencia June 3-6, 2013 14 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 If this happens for every weakly compact polynomial, we say that X has the polynomial Daugavet property. Theorem Suppose that A is a uniform algebra whose Choquet boundary has no isolated points. For every P S (AX ),f0 SAX and ✏ > 0,thereexist 2 P 2 some ! S and g B X such that Re!P(g) > 1 ✏ and 2 C 2 A f + !g > 2 ✏. k 0 k

Polynomial Daugavet Property

ABanachspaceX is said to have the Daugavet property if the norm identity, so called the Daugavet equation,

Id + T =1+ T k k k k holds for every rank-one operator (and hence for every weakly compact operator) T L(X ). 2

Worksop on Functional Analysis Valencia June 3-6, 2013 15 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Theorem Suppose that A is a uniform algebra whose Choquet boundary has no isolated points. For every P S (AX ),f0 SAX and ✏ > 0,thereexist 2 P 2 some ! S and g B X such that Re!P(g) > 1 ✏ and 2 C 2 A f + !g > 2 ✏. k 0 k

Polynomial Daugavet Property

ABanachspaceX is said to have the Daugavet property if the norm identity, so called the Daugavet equation,

Id + T =1+ T k k k k holds for every rank-one operator (and hence for every weakly compact operator) T L(X ). 2 If this happens for every weakly compact polynomial, we say that X has the polynomial Daugavet property. Theorem Suppose that A is a uniform algebra whose Choquet boundary has no isolated points. For every P S (AX ),f0 SAX and ✏ > 0,thereexist 2 P 2 some ! S and g B X such that Re!P(g) > 1 ✏ and 2 C 2 A f + !g > 2 ✏. k 0 k

Worksop on Functional Analysis Valencia June 3-6, 2013 15 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Since given a ball U in Cn or the polydisk Dn the Choquet boundary of (U) does not have isolated points, A Corollary If U is a ball in Cn or the polydisk Dn,then (U, X ) has the polynomial A Daugavet property for every complex Banach space X. In particular, (U) A has the polynomial Daugavet property.

Polynomial Daugavet Property

Corollary If A is a uniform algebra whose Choquet boundary has no isolated points, then every weakly compact P (AX ; AX ) satisﬁes the Daugavet 2 P equation which means that AX has the polynomial Daugavet property.

Worksop on Functional Analysis Valencia June 3-6, 2013 16 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Polynomial Daugavet Property

Since given a ball U in Cn or the polydisk Dn the Choquet boundary of (U) does not have isolated points, A Corollary If U is a ball in Cn or the polydisk Dn,then (U, X ) has the polynomial A Daugavet property for every complex Banach space X. In particular, (U) A has the polynomial Daugavet property.

S = S(B , x⇤, ✏)= x B : Rex⇤(x) > 1 ✏ , x⇤ S X { 2 X } 2 X ⇤ such that x S and dist(y, aconv(S)) < ✏. 2

Lushness

The concept of lushness was introduced to characterize an inﬁnite dimensional Banach space with the numerical index 1. The lushness has been known to be the weakest among quite a few isometric properties in the literature which are sucient conditions for a Banach space to have the numerical index 1.

Worksop on Functional Analysis Valencia June 3-6, 2013 17 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Lushness

S = S(B , x⇤, ✏)= x B : Rex⇤(x) > 1 ✏ , x⇤ S X { 2 X } 2 X ⇤ such that x S and dist(y, aconv(S)) < ✏. 2

Worksop on Functional Analysis Valencia June 3-6, 2013 17 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Corollary If X is lush, then (B ) is lush for any Banach space Y . Au Y

Lushness

Theorem Suppose that A is a uniform algebra. Then X is lush if and only if AX is lush. In particular, A is lush.

Worksop on Functional Analysis Valencia June 3-6, 2013 18 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Lushness

Theorem Suppose that A is a uniform algebra. Then X is lush if and only if AX is lush. In particular, A is lush.

Corollary If X is lush, then (B ) is lush for any Banach space Y . Au Y

Worksop on Functional Analysis Valencia June 3-6, 2013 18 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 This property was introduced to characterize a Banach space X such that the pair (`1, X ) has the Bishop-Phelps-Bollob´as property for operators.

AHSP

ABanachspaceXissaidtohavetheAHSP if for every ✏ > 0 there exist (✏) > 0and⌘(✏) > 0 with lim✏ 0+ (✏)=0suchthatforeverysequence ! (x ) B and for every convex series 1 ↵ satisfying k k1=1 ⇢ X k=1 k

1 P ↵ x > 1 ⌘(✏) k k k=1 X there exist a subset A N,asubset zk : k A SX and x⇤ SX ⇢ { 2 } ⇢ 2 ⇤ such that (i) ↵ > 1 (✏) k A k (ii) z 2 x < ✏ for all k A,and Pk k k k 2 (iii) x (z )=1forallk A. ⇤ k 2

Worksop on Functional Analysis Valencia June 3-6, 2013 19 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 AHSP

1 P ↵ x > 1 ⌘(✏) k k k=1 X there exist a subset A N,asubset zk : k A SX and x⇤ SX ⇢ { 2 } ⇢ 2 ⇤ such that (i) ↵ > 1 (✏) k A k (ii) z 2 x < ✏ for all k A,and Pk k k k 2 (iii) x (z )=1forallk A. ⇤ k 2 This property was introduced to characterize a Banach space X such that the pair (`1, X ) has the Bishop-Phelps-Bollob´as property for operators.

Worksop on Functional Analysis Valencia June 3-6, 2013 19 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Theorem Suppose that A is a uniform algebra. Then X has the AHSP if and only if AX has the AHSP.

Corollary If X has the AHSP, then (B ) has the AHSP for any Banach space Y . Au Y

AHSP

Since every lush space has the AHSP, We have the following. Corollary Suppose that A is a uniform algebra. 1 If X is lush, then AX has the AHSP. In particular, A has the AHSP. 2 If AX is lush, then X has the AHSP.

Worksop on Functional Analysis Valencia June 3-6, 2013 20 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Corollary If X has the AHSP, then (B ) has the AHSP for any Banach space Y . Au Y

AHSP

Theorem Suppose that A is a uniform algebra. Then X has the AHSP if and only if AX has the AHSP.

Worksop on Functional Analysis Valencia June 3-6, 2013 20 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 AHSP

Theorem Suppose that A is a uniform algebra. Then X has the AHSP if and only if AX has the AHSP.

Corollary If X has the AHSP, then (B ) has the AHSP for any Banach space Y . Au Y

Worksop on Functional Analysis Valencia June 3-6, 2013 20 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20