As for extending real-valued continuous functions or continuous pseudometrics on a subspace to the whole space, notions of z-, C * -, C-, P -and P γ -embeddings are known. As a cardinal generalization of z-embedding, Blair defined in 1985 the notion of z γ -embedding with γ ω, where z ω -embedding coincides with z-embedding. On the other hand, since P ω -embedding equals C-embedding, P γ -embedding can be also regarded as a cardinal generalization of C-embedding.
Recently Ohta asked if a cardinal generalization of C * -embedding can be defined so that this property plus U ω -embedding is equal to P γ -embedding, and it is itself equals C * -embedding in case γ = ω. In this paper, we give a cardinal generalization of C * -embedding, called (P * ) γ -embedding, and answer this problem. As a characterization of (P * ) γ -embedding, we show that (P * ) γ -embedding naturally admits its description by using continuous maps from a subspace into the hedgehog with γ spines. We also give a new extension-like property called weak z γ -embedding, with which z-(respectively C * -, C-or U ω -) embedding equals z γ -(respectively (P * ) γ -, P γ -or U γ -) embedding.