Generalized $q$-Gaussian von Neumann algebras with coefficients, III. Unique prime factorization results

Abstract: We prove some unique prime factorization results for tensor products of type

I I_{1}

factors of the form

G a m m a_{q} (m a t h b b C, S o t i m e s H)

arising from symmetric independent copies with sub-exponential dimensions of the spaces

D_{k} (S)

and dim

(H)

finite and greater than a constant depending on

q

.