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Unitary Operators in C-Algebras

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Unitary Operators in C-Algebras Pacific Journal of Mathematics UNITARY OPERATORS IN C∗-ALGEBRAS JAMES G. GLIMM AND RICHARD VINCENT KADISON Vol. 10, No. 2 October 1960 UNITARY OPERATORS IN C*-ALGEBRAS JAMES G. GLIMM AND RICHARD V. KADISON 1. Introduction. We present several results concerning unitary operators in uniformly closed self-ad joint algebras of operators on a Hu- bert space (C*-algebras). Section 2 contains these results the key one of which (Theorem 1) asserts a form of transitivity for unitary operators in an irreducible C*-algebra (an application of [2, Theorem 1]). Section 3 consists of some applications. The first (Corollary 8) is a clarification of the relation between unitary equivalence of pure states of a C-*algebra and of the representations they induce. The most desirable situation prevails: two pure states of a C*-algebra are unitarily equivalent (i.e. conjugate via a unitary operator in the algebra) if and only if the re- presentations they induce are unitarily equivalent. The second application (Corollary 9) provides a sufficient condition for two pure states p and τ to be unitarily equivalent: viz. || p — τ || < 2. The final application (Theorem 11) is to the affirmative solution of the conjecture that the * operation is isometric in J5*-algebras [1], We use the notation o(A) for the spectrum oί A; C for the set of complex numbers of modulus 1; .5/ ~ for the strong closure of the set of operators £f\ and ωx for the state, A —> (Ax, x)9 due to the unit vector x. Our C*-algebras all contain the identity operator I. 2 Unitary operators* The theorem which follows establishes an n-folά transitivity property for the unitary operators in an irreducible C*-algebra. Its relation to [2, Theorem 1] is clear—it is, in fact, the multiplicative version of the self-ad joint portion of that theorem. THEOREM 1. If 21 is a C*-algebra acting irreducibly on 3ίf and V is a unitary operator on Sff such that Vxk = yk, k = 1, , n, then there is a unitary operator U m 21 such that Uxk = yk and σ( U) Φ C. Proof. Passing to an orthonormal basis for the finite-dimensional space generated by {x19 , xn}, we see that there is no loss in generality if we assume that {xlf •••,#„}, and hence {y19 , yn}, are orthonormal sets. Moreover, employing a unitary extension of the mapping carrying Xj onto yj9 j = 1, , n to the space generated by {x19y19 , xn9 yn} and a diagonalizing basis for this unitary operator; we see that it suffices to consider the case in which Vx5 = βjXjf | βj \ = 1, j = 1, , n. Choose real a5 in the half-open interval ( — π, π] such that expia3 = βj9 and let A be a self-ad joint operator in 2ί such that Ax5 = a5x3 (such Received April 27, 1959. The second author is an Alfred P. Sloan Fellow. 547 548 JAMES G. GLIMM AND RICHARD V. KADISON an operator exists by [2, Theorem 1]). Define g(a) as a for a in [min {aό}, max {aj}], as min {a5} and max {a3}, for a < min {a3} and α > max^}, respectively. Then g(A) is a self-ad joint operator in 2ί with spectrum in [min {oc3}, max {#?}], and #04.)#j = otjXj. It follows that exp ig(A) is a unitary operator U in 21, tf(ί7) Φ C, and ί/a^ = βjXj. Another unitary analogue of a known result which seems of some value is the following variant of Kaplansky's Density Theorem [3, Theorem 1], It is a consequence of Kaplansky's theorem and some commutative spectral theory. THEOREM 2. If ^(21, k) is the set of unitary operators in the C*- algebra 21 whose distance from I does not exceed k, then ^(21, k)~ con- tains ^(?X~, k). Proof. Note that || U — I\\ < k, for a unitary operator U, if and only if σ{U) is contained in {z: \ z — 11 < k, \ z \ = 1}, a closed subset Sk of the unit circle C. From spectral theory, each unitary operator is a uniform limit of unitary operators which are finite linear combinations of orthogonal spectral projections for it, and which do not have —1 in their spectra (i. e. whose distance from / is less than 2). Thus, it suf- fices to consider the case where k < 2. Assuming k < 2, let arg z be that number in the open interval ( — 7Γ, π) such that z — exp [i arg z], for z in Sk; and let/ be a continuous extension of arg to C. If a — 2 sin"1(fc/2) and ,^(2I, a) denotes the set of self-ad joint operators in 21 with norm not exceeding α, then / maps ^(21, k) into ,^(2I, a) continuously in the strong topology, C/=exp [if (17)], and exp maps i^(2ϊ, a) into ^/(2ί, fc) continuously in the strong topology, from spectral theory, [3, Lemma 3], and [3, Lemma 2], Thus, if U lies in ^(2Γ, k),f(U) lies in ^(2X~, α) and is a strong limit point of ^(21, α), from [3, Theorem 1]; so that U(= exp [if (U)]) is a strong limit point of ^(St, fc). In the next lemma, we make use of Mackey's concept of disjoint representations [5]. These are ^representations of self-ad joint operator algebras which have no unitarily equivalent non-zero subrepresentations (the restriction of the representation to an invariant subspace). The application in [5] is to unitary representations of groups and ours is to ^-representations of algebras—the difference is slight, however; and our lemma and proof are valid for groups. LEMMA 3. // {φa} are *-representations of the self-adjoint operator algebra 21, then {φa} consists of mutually disjoint representations if and only if φ(21)" = Σ Θ (Φ*(2l)-), where φ= and s the Proof. Suppose φΛ{%) acts on ^,T=Σθ^ P« * UNITARY OPERATORS IN C*-ALGEBRAS 549 orthogonal projection of ^ upon 3%. If Φ(2l)- = Σ Θ (Φ (3I)~), and t/is a partial isometry [6] of Ea{Sίζ) onto E„,(£%,), where α^α' and #„, £7Λ, are projections commuting with φΛ(5I) and </v(3I), respectively, such that Uφa(A)U* = φa,(A)Ea, for each A in 2ί, then Z7commutes with φ(2ί). In fact, ϊ/φ(il)=ϊ7φΛ(A)=φΛ,(A)J7=φ(A)ί7. Thus Ucommutes with Σ&ΦM) and, in particular, with each Pa. But UP# = Z7 = PJJ = 0; so that 0 = ϋ7Λ = 2?Λ/ and {φ*} consists of mutually disjoint representations. If the φa are mutually disjoint and V is a partial isometry in the commutant of φ(SX) with the initial space EΛ in Pa and final space 2?Λ, in P«, (cf. [6]); then VEΛφa{A)EaV* = FF*Fφ(A)F*FF*=#Λ,φ(A)FF*#Λ, - φΛ,(A)EΛ,, for each A in 2ί. Thus, by disjointness, 2£Λ and 2£Λ/ are 0. It follows that the central carrier of Pa is orthogonal to that of each PΛ,, and hence to each PΛ,, with a' Φ a (see [4], for example). Since ΣΛ = ^ and the central carrier of P* contains Pa, PΛ is its own central carrier. In particular, Pa lies in the center of the commutant and there- fore in φ(9ϊ)-. It is immediate from this that φ(2I)~ = Σ Θ (Φ«(2t)~). Since the commutant of an irreducible representation consists of scalars, two such are either unitarily equivalent or disjoint. From this and Lemma 3, we have as an immediate consequence: COROLLARY 4. If {φa} is a family of irreducible *-representations of a self-adjoint operator algebra SI, no two of which are unitarily equivalent, then φ(SI)~ = Σθ ^L> where φ = Σ 0 Φ* and &» is the al- gebra of all bounded operators on the representation space of φa. We shall need a result asserting the possibility of "lifting" unitary operators from a representing algebra to the original algebra under cer- tain circumstances. LEMMA 5. If φ is a *-representation of the C*-algebra 21 and ϋ is a unitary operator in Φ(Ά) with o{U) Φ C, there is a unitary ope- rator Uo in 31 such that φ(U0) = U. Proof. As in Theorem 2, we can find a continuous function f on C such that f(U) is self-ad joint and exp [if (U)] = U. Let A be a self- ad joint operator in 21 such that φ(A) = f{U). (Recall that φ(2I) is a C*- algebra, so that/(E/) lies in φ(2ί). If φ(B) =f(U) then A may be chosen as (£* + B)I2.) If Uo = exp iA then φ(UQ) = exp [if (17)] = Z7, by uni- form continuity of φ. REMARK 6. It may not be possible to lift a given unitary operator (as indicated by the condition σ(U)ΦC in Lemma 5). In fact, illustrat- ing this with commutative C*-algebras, we may deal with the algebras of continuous complex-valued functions on compact Hausdorff spaces and 550 JAMES G. GLIMM AND RICHARD V. KADISON unitary functions on them (functions with modulus 1). View C as the equator of a two-sphere S9 and let a be the inclusion mapping of C into S. Then a induces a homomorphism of the function algebra of S onto that of C ("onto" by the Tietze Extension Theorem) which is, of course, the mapping that restricts a function on S to C. The identity map- ping of C onto C is a unitary function on C which does not have a continuous unitary extension to S; for such an extension restricted to one hemisphere would amount to a retraction of the disk onto its boundary.
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