Group algebras of vector-valued functions

Symmetry of Generalized Group Algebras

“…The proof of the first of these implications will be accomplished by showing that if tEB"(G, A)QB1(G, A) then the spectrum of t in BP(G, A), ap(t) equals the spectrum of t in B1(G, A), ai(t). Since BP(G, A)QB1(G, A), clearly o-i7)Co-P(0-On the other hand, BP(G, A) is an ideal in B^G, A) [6]. Recall that 0^X$cri7) iff t/\ has a quasi- Note that since Ai is finite-dimensional the algebraic tensor product normed with the greatest cross norm is complete.…”

Symmetry of generalized group algebras

“…The proof of the first of these implications will be accomplished by showing that if tEB"(G, A)QB1(G, A) then the spectrum of t in BP(G, A), ap(t) equals the spectrum of t in B1(G, A), ai(t). Since BP(G, A)QB1(G, A), clearly o-i7)Co-P(0-On the other hand, BP(G, A) is an ideal in B^G, A) [6]. Recall that 0^X$cri7) iff t/\ has a quasi- From this lemma it follows that if B1(G, A) is symmetric and tEB"(G, A), then -l$<Ti(t*t) and therefore -l(£ap(t*t).…”

“…The proof given here depends on the minimal ideal structure of L1(G) via the identification B1(G, A) = L1(G)®yA [6], based on a result by Grothendieck [l, p. 59]. We present the proof as a sequence of lemmas.…”

On Group Von Neumann Algebras with Vector-Valued Functions

“…These spaces form Banach algebras under usual operations and convolution. In [13], D. Z. Spicer extended the group algebras ( )and ( ) to group algebras of vectorvalued functions respectively denoted ( , )and ( , ). Mainly, ( , ) is the space of all continuous functions : → such that ∫ ‖ ( )‖ < ∞ (usually denoted (G, A))), and ( , ) is the space of all continuous functions from to , where is a Banach algebra.…”